Modeling magmatic accumulations in the upper crust: Metamorphic implications for the country rock
Madison M. Douglas, Adelina Geyer, Antonio M. Álvarez-Valero, Joan
Martı́
PII:
DOI:
Reference:
S0377-0273(16)30018-X
doi: 10.1016/j.jvolgeores.2016.03.008
VOLGEO 5788
To appear in:
Journal of Volcanology and Geothermal Research
Received date:
Revised date:
Accepted date:
23 October 2015
29 February 2016
8 March 2016
Please cite this article as: Douglas, Madison M., Geyer, Adelina, Álvarez-Valero, Antonio
M., Martı́, Joan, Modeling magmatic accumulations in the upper crust: Metamorphic
implications for the country rock, Journal of Volcanology and Geothermal Research (2016),
doi: 10.1016/j.jvolgeores.2016.03.008
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Corresponding author: Adelina Geyer, Group of Volcanology, SIMGEO (UB-CSIC),
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Institute of Earth Sciences Jaume Almera, ICTJA-CSIC, Lluis Solé i Sabaris s/n, 08028
Barcelona, Spain. ([email protected])
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The extent and degree of the generated
aureole depend on many parameters, including: temperature, composition and thermal
properties (i.e. thermal conductivity, specific heat capacity) of the magma and country
rock; the degree of crystallization of the magma; the size and geometry of the reservoir;
the latent heat of fusion for the crystallizing magma; and the amount of heat energy
consumed by endothermic metamorphic reactions taking place in the aureole (Johnson et
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al., 2011;Peacock and Spear, 2013). Additionally, at shallow crustal levels, hydrothermal
processes may play an important role in incorporating cations and H2O into the system,
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as well as redistributing the heat released from the cooling magma reservoir that can
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affect contact metamorphism equilibria (e.g. Peacock and Spear, 2013).
, 2013
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Heat transfer models
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The temperature distribution within the magma chamber and the host rock is
calculated using FEM, by solving the pure conductive heat transfer equation, and
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assuming the effects of viscous heating and pressure-volume work are negligible (see
[1]
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Rodríguez et al., 2015 for further details):
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where the equation parameters refer to density (), specific heat capacity at constant
pressure (Cp), temperature (T), time (t), thermal conductivity (k) and Q denotes heat
sources other than viscous heating (see Table 1 for details of the thermal and physical
parameters).
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Symbol
AD
Cp
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d
Supp. Mat. Table S8
Supp. Mat. Tables S4-6
Supp. Mat. Tables S4-6
Specific heat capacity of crust4
J/(kg K)
Specific heat capacity of magma (melt phase)
Specific heat capacity of magma (solid phase)
J/(kg K)
J/(kg K)
Supp. Mat. Table S7
Magma chamber depth
km
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Cpmelt
Cpsolid
0.25-2
e
EA
120,000
magma
thermal_grad
T_liquidus
T_solidus
T_surface
T0_magma
T0_crust
P

J/mol
Magma chamber height
km
m2/s
m2/s
 · Cp · 
Thermal conductivity
W/(m K)
Latent heat of crystallization1
Melt fraction3
Subcrustal heat flow1
Surface heat flow1
Crust density
Magma density3
Solid fraction3
kJ/kg
W/m2
W/m2
kg/m3
kg/m3
-
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Thermal diffusivity of crust4
400
Supp. Mat. Tables S1-3
0.03
0.07
2700
2344-2694
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
R
Activation energy
Supp. Mat. Table S8
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k
L
ϕ
-
Thermal diffusivity of magma 4
crust
crust
magma
Magma chamber aspect ratio
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1.061-9.141
h
Qmantle
Qsurface
SI Unit
Pa s
J/(kg K)
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Cpcrust
Variable
Dorn parameter
Specific heat capacity at constant pressure
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Value
109
1- 
8.314
25-30
Supp. Mat. Tables S1-3
Supp. Mat. Tables S1-3
298.15
1-3106
Eq. 3
Molar gas constant
Crustal thermal gradient
Magma liquidus temperature3
Magma solidus temperature3
Surface temperature
Initial magma temperature
Initial crustal temperature
Magma chamber internal pressure at h/2
Country rock viscosity
J/(mol K)
K/km
K
K
K
K
K
Pa
Pa s
4.243-19.696
Magma chamber width
w
km
1, Bea (2010); 2, Whittington et al. (2009); 3, use Rhyolite MELTS code by Gualda et al. (2012); 4 Rodríguez
et al. (2015)
Physical and thermal parameters
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We consider the modeled magma chamber to be emplaced immediately before it
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begins cooling. The completely liquefied magma reservoir (Gelman et al., 2013) is
assumed to intrude in a single episode without further replenishment (Gutiérrez and
carried
out
with
COMSOL
Multiphysics
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are
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Parada, 2010). The geometric modeling, mesh discretization and numerical computations
v4.4
software
package
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(http://www.comsol.com). To simulate the solidifying magma, we use the heat transfer
with phase change module, which allows us to solve Equation 1 after setting the
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properties of a phase-change material (from liquid to solid) according to the Apparent
Heat Capacity formulation (AHC) (see Rodríguez et al., 2015 and references therein).
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The AHC formulation is considered to be the best representation of a naturally occurring
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wide phase-change temperature interval, as occurs during magma cooling (Rodríguez et
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al., 2015). The latent heat of crystallization is accounted for by increasing the heat
capacity of the material within the phase change temperature range. For further details
on the methodology, the reader is referred to the work by Rodríguez et al. (2015). A
comprehensive report detailing one of the models, which includes all settings within
COMSOL (e.g. model properties; physics settings; geometry; mesh), as well as an
example of one of the models, are available from the corresponding author upon request
(COMSOL Multiphysics commercial software V4.4 or greater and heat transfer with
phase change module are required).
The performed FE models are axisymmetric and constructed over a cylindrical
coordinate system with positive z values related to altitudes above sea level (Fig. 1a). The
magma chamber geometry is oblate, spherical or prolate in shape with height h and width
w (Fig. 1a). The computational domain corresponds to a section of the crust with a 40 km
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radius stretching to a depth of 40 km below sea level (Fig. 1a). The FE mesh consists of
between 200,000 and 400,000 linear triangular elements up to 250 m in size farther from
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the magma reservoir, and 2 m in size near the edge of the magma chamber (Fig. 1b).
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Time steps taken to perform the calculations range from less than a year at the beginning
of the cooling processes, up to 300 years to speed calculations once reached the
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metamorphic peak. We have tested that the chosen time steps do not affect the results
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obtained.
The selected starting magmatic compositions for the numerical simulations
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correspond to three representative types that range from alkaline basalt to rhyolite (Table
2). The former, typical of intraplate magmatism, is taken from Polat (2009). The
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intermediate magma is of andesitic composition and from a subduction zone taken from
Yogodzinski et al. (1994). The rhyolitic composition is from an archetypal case of pelitic
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crustal melting taken from Sensarna et al. (2004). Depending on the model, pressure at
the magma chamber center ranges from 1 to 3 kbar (Supplementary Material, Table S1).
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Andesite 35-10/3 [1]
Rhyolite DM 4 [2]
Alkaline Basalt SC95-86 [3]
SiO2
56.88
75
50
TiO2
0.68
0.29
2.2
Al2O3
18.36
11.69
FeO*
4.72
2.8
MgO
4.04
MnO
0.1
CaO
7.48
Na2O
4.05
K2O
1.3
P2O5
0.11
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4.9
0.03
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0.3
0.79
9.6
2.2
3.2
5.11
0.04
0.4
0.4
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[2] Sensarma et al., 2004
[3] Polat, 2009
13.95
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[1] Yogodzinski et al., 1994
13.5
0.15
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* FeO as total iron
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Major elements (wt%)
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Table 2: Starting magma compositions for the different numerical simulations
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The melt () and solid (ϕ) fractions, as well as the thermal properties of the
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crystallizing magmas, are determined using the Rhyolite-MELTS code (Gualda et al.,
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2012). Results are reported in the Supplementary Material (Table S2 – S7) and are used
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as input data for the models. We simulate isobaric cooling at 1, 2 and 3 kbar pressure,
with redox conditions one log unit above the quartz-fayalite-magnetite (QFM) oxygen
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buffer.
Using the thermal properties provided by MELTS, we explicitly account for the
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temperature dependence of thermal diffusivity () and heat capacity (CP). Incorporating a
temperature-dependent diffusivity is critical due to its strong influence on the
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temperature-dependence of thermal conductivity (k= ρ · CP ·) at high temperatures
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(Nabelek et al., 2012). Average magma density magma values obtained from MELTS are
2344 kg/m3, 2440 kg/m3 and 2694kg/m3 for the rhyolitic, andesitic and basaltic magmas,
respectively. For most models
, we consider a crustal
density of 2700 kg/m3 (Whittington et al., 2009), a temperature-dependent thermal
diffusivity as provided by Chapman and Furlong (1992) for the upper crust and a specific
heat capacity calculated with Equations 3 and 4 of Whittington et al. (2009)
. In order to evaluate the influence of the crustal
thermal properties on the results obtained, we have also considered for specific models
alternative temperature-dependent thermal diffusivity and specific heat capacity for the
country rock
. The latter are further described in Table
S8 of the Supplementary Material.
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The input temperature profile for the country rock is a typical geothermal gradient
varying from 25 to 30ºC/km depending on the model
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. We assume a relatively uniform lateral temperature
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profile, and consider the surrounding walls of the country rock to have no heat flux (i.e.
the outer edge of the rotationally symmetric model is insulating). The subcrustal mantle
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heat flow Qmantle = 0.03 W/m2 and the surface heat flow Qsurface = 0.07 W/m2 are assigned
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to the bottom and top limits of the computational domain, respectively (Fig. 1a) (Table
1). For the magma chamber, we assume an initial temperature T0_magma equivalent to the
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liquidus temperature of the magma (Supplementary Material, Tables S2-S4).
Following the approach by Jellinek and DePaolo (2003) and Chen and Jin (2011),
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under high temperatures and confining pressures, crustal rocks are assumed to behave
viscoplastically or also referred to as a solid-state creep flow. Thus, the host rock
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rheology can be described by the Weertman-Dorn power-law formulation where the
relationship between strain rate and deviatoric stress is defined as follows (Jellinek and
DePaolo, 2003 and references therein; Chen and Jin, 2011):
[2]
where K is a constant that depends on the material, G is the activation energy for creep, n
is the power law exponent, v is the deviatoric stress, and R is the molar gas constant
(Table 1). Interesting feature of this approach to the knowledge of the wall-rock rheology
is that the effective wall-rock viscosity is an explicit function of temperature and strain
rate (Jellinek and DePaolo, 2003). The thermal effects on the country rock are examined
as a function of the temperature-dependent viscosity determined by the temperature field
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distribution and calculated using the Arrhenius formulation (Del Negro et al., 2009;
[3]
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Chen and Jin, 2011; Gregg et al., 2012) :
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where AD is the Dorn parameter (a material constant) and EA is the activation energy
(Table 1). According to Equation 2, higher temperatures close to the magma chamber-
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host rock boundary would implicate lower viscosities. Thus, the viscosity values of the
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continental crust (i.e. > 1019 Pa s).
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heated aureole are expected to be much lower than the average viscosity estimates for
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2.2 Mineral assemblage prediction: phase diagram modeling
Magma chamber emplacement disrupts the temperature-dependent chemical
equilibrium of the surrounding crust, triggering reactions to establish a new equilibrium.
Thereby, we use thermodynamic modeling (P-T pseudosections) to predict equilibrium
phase assemblages, reactions and modal changes occurring in the country rock around the
magmatic body.
We use the thermodynamic database of Holland and Powell (1998, with updates) and
Perple_X (Connolly, 2005) in the system Na2O-CaO-K2O-FeO-MgO-Al2O3-SiO2-H2O
(NCKFMASH) (Fig. 2, Supplementary Material Figures S1 and S2). We consider three
representative upper continental crustal compositions to simulate the magma
emplacements effects on country rock, namely (Fig. 2, Supplementary Material Figures
S1 and S2): (i) the anhydrous standard upper continental crust of Rudnick and Gao
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(2003); (ii) a typical hydrous metapelite (Tinkham et al., 2001); and (iii) a metamarl
(Miron et al., 2013). The growth of hydrated phases such biotite and cordierite in the
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standard continental crust requires a minimum amount of H2O. Therefore, we include
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0.25 of wt% water in the bulk composition of Rudnick and Gao (2003) to permit the
formation of hydrated phases.
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The activity–composition models for silicate melt (M), garnet (Grt) and biotite (Bt) are
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from White et al. (2007); staurolite (St), epidote (Ep) and cordierite (Crd) are taken from
Holland and Powell (1998); plagioclase (Pl) and K-feldspar (Kfs) are from Holland
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(2003); calcite (Cc) is from Anovitz and Essene (1987); chlorite (Chl) is from Holland et
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al. (1998); and white mica (Ms, Pa, Ma) is from Coggon and Holland (2002).
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Figure 2 (1.5 column): Pressure-Temperature pseudosection in the NCKFMASH system.
Darker shading indicates higher variance. Bulk composition (wt.% normalized to 100) is
of a hydrated metapelite from Tinkham et al. (2001): Na2O: 1.65, CaO: 1.21, K2O: 3.70,
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FeO: 6.87, MgO: 3.44, Al2O3: 16.88, SiO2: 60.78, H2O: 5.44. And (andalusite), St
(staurolite); Bt (biotite); Ms (muscovite); Ep (epidote); Pa (paragonite); Pl
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(plagioclase); Qtz (quartz); M (melt); Crd (cordierite); Grt (garnet); Chl (chlorite); Kfs
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(K-feldspar).

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[4]
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
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
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Figure 3 (double column): Summary of the most relevant results discussed in the text
regarding temperature distribution at the country rock due to the heated magma
chamber. (a) Temperature distribution along the D-D’ profile at different time steps.
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Results
for
the
oblate
(left)
(M_reference
model)
and
spherical
(right)
(M_shape_spherical model) reservoir are illustrated. Depth z is normalized towards the
magma chamber center depth d, i.e. z/depth = -1 indicates magma chamber horizontal
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axis. Temperature increase T along the D-D’ (b) and E-E’ (c) profiles for the oblate
reservoir (M_reference model). Comparison of the temperature evolution at the three
reservoirs
(M_reference,
M_shape_sphere
and
M_shape_prolate
models,
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(e)
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control points (A-C) for the case of the oblate and spherical (d) and oblate and prolate
respectively). Model name as listed in Table S1 of the Supplementary Material indicated
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within brackets. Roman numerals detailed in the text.

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ν

ν
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ν
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Figure 4 (double column): Summary of the most relevant results discussed in the text
regarding viscosity changes at the country rock due to the magma chamber
accumulation. (a) Viscosity of the country rock at the metamorphic peak of point C for
the reference model (M_reference, i.e. rhyolitic magma at 3 kbar depth). (B) Country
rock viscosity along the E-E’ crustal profile at different time steps for the reference
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oblate reservoir. Evolution of the country rock’s viscosity at the three control points (A,
B, C) for the oblate (c), spherical (d) and prolate (e) reservoirs (M_reference,
M_shape_sphere and M_shape_prolate models, respectively). The dashed lines with
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double arrows indicate the time interval in which the control crustal points remain below
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viscosity value of 1016 Pa s.
III) and spherical (Fig. 5b, label IV) reservoirs, respectively.
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Figure 5 (double column): (a) Evolution of the metamorphic aureole for the reference
model at different time steps corresponding to the metamorphic peak of the three control
points: A, B and C. (b) Predicted mineral assemblage for the case of a spherical
reservoir. The time step corresponds to the metamorphic peak of point C (i.e. magma
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chamber bottom). (c) The same for the prolate reservoir at t= 0s and at the metamorphic
peak of C (i.e. magma chamber bottom). (d) For illustrative purposes we also show the
predicted mineral assemblage for the metapelitic crust color-coded for the different
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phases identified in the results shown in (a), (b) and (c). The green, red and blue
horizontal lines indicate the temperature increase experienced at the top (control point
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A), horizontal axis (control point B) and bottom (control point C) of the magma chamber
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in the case of the spherical (dotted lines), oblate (solid lines) and prolate (dashed lines)
reservoirs. Roman numerals are detailed in the main text. And (andalusite), St
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(staurolite); Bt (biotite); Ms (muscovite); Ep (epidote); Pa (paragonite); Pl
(plagioclase); Qtz (quartz); M (melt); Crd (cordierite); Grt (garnet); Chl (chlorite); Kfs
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(K-feldspar).
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We also run the same simulations for two other crustal compositions including: (i) a
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crustal metamarl (Supplementary Material, Fig. S2), which shows a similar metamorphic
trend to the described above for the metapelitic case but with different mineral
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assemblages; (ii) a typical upper crust (Rudnick and Gao, 2003), which shows no contact
metamorphic changes as the P-T evolution of the three control points are stable within the
same equilibrium mineral assemblage (Supplementary Material Fig. S1).
Numerical modeling of heat transfer from the magma chamber to the surrounding
crust
There is a general agreement when computing numerical simulations of cooling
evolution in a magma chamber, about the restrictions or simplifications that may
potentially influence the estimates of the thermal effects in the surrounding host rock. On
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this regard, many of the existing numerical models so far: (i) simulate the reservoir as a
pressurized cavity with a constant heat flux to the country rock (e.g. Jellinek and De
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Paolo, 2003); (ii) assume the thermal properties to keep constant values, including
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diffusivity and specific heat capacity for a constant P, regardless of T (e.g. Gottsmann
and Odbert, 2014); (iii) ignore the existence of hydrothermal fluids in the country rock
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(e.g. Gregg et al., 2012); (iv) consider just the conductive term of the heat transfer
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equation assuming that no convection is taking place inside the magma chamber (e.g.
Gelman et al., 2013); (v) consider the magma reservoir to be instantaneously emplaced
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(e.g. Gutiérrez and Parada, 2010); and (vi) ignore endothermic metamorphic reactions
occurring in the country rocks during the thermal evolution of the aureole (e.g. Jellinek
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and De Paolo, 2003).
Concerning the first simplification, i.e. considering constant heat flux from the
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magma chamber uniformly distributed along the reservoir-host rock contact, it leads to an
overestimation of the thermal effects at the country rock (e.g. Gregg et al. 2012). Yet the
heat flux from the magma chamber to the country rock is high at the beginning of the
magma-cooling episode, but rapidly decreases as the reservoir cools down (Fig. 6).
Hence, keeping the thermal heat flux constant during the simulation leads to an
overheating of the country rock over long periods of time, which translates in to an
overestimate of the thermal aureole’s size and duration.
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Figure 6 (single column): Heat flux along the D-D’ profile at 5 m from the magma
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chamber – host rock contact for different time steps. Depth z is normalized towards the
magma chamber center depth d, i.e. z/depth = -1 indicates magma chamber horizontal
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axis. Results correspond to those obtained with the reference model (M_reference).
The second assumption of temperature-independent thermal properties of the
magma and country rock limits the model’s accuracy in describing thermal evolution
over large temperature ranges. The numerical simulations presented by Nabelek et al.
(2012) demonstrated that the modeled longevities of magmatic systems are significantly
extended by accounting for thermal dependencies in material properties. The time for the
magma solidification is long because of both its low thermal diffusivity and the wall
rocks become more insulating as temperature rises (Whittington et al., 2009,
Supplementary Material Table S8). Thus, the temperature increase rate and the duration
of elevated temperature regimes in the country rock are affected. According to Nabelek et
al. (2012), assuming constant diffusivity values does not significantly impact peak
temperature values at any location in the country rock. However, once the country rocks
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are heated up and their thermal diffusivity decreases, they remain at high temperatures for
longer than in models assuming a constant thermal diffusivity (Nabelek et al., 2012).
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As mentioned in the methodology section, our models overcome the first two
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restrictions. The temperature-dependent thermal properties of the crust are considered to
be homogeneous among the whole computational domain (i.e. crustal section). However,
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results obtained when changing the thermal properties of the crust (Supplementary
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Material Table S8) indicate that variations in the thermal diffusivity and specific heat
capacity may influence the thermal evolution of the country rock, especially the
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temperature value at the metamorphic peak (Supplementary Material Figure S4).
Therefore, when working with real case studies, the implementation of a proper
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numerical model considering the individual geologic units and their specific temperaturedependent thermal properties is strongly recommended (e.g. Cook and Bowman, 1994;
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Cui et al. 2001, Álvarez-Valero et al., 2015; Pla and Álvarez-Valero, 2015). This would
allow accounting for variations of the thermal properties due to changes of the host rock
composition, water content, porosity, etc.
Regarding the third and fourth limitations, i.e. not including magma convection
within the chamber as well as neglecting the presence of fluids in the country rock, their
effects on the outcomes are still under discussion. On the one hand, internal magmatic
convection has been shown to affect peak temperatures in the contact aureole (Jaeger,
1964;Cook and Bowman, 1994). These effects are expected to be minor on distance
profiles normal to the igneous contacts of planar or regular geometries, except when
being either close to the reservoir-rock contact (Bowers et al., 1990), or next to local
reservoir irregularities, such as re-entrants and apophyses (Cook and Bowman, 1994).
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Bowers et al. (1990) demonstrated that magmatic convection can also result in isotherm
geometries that significantly differ from pure conduction, causing isotherms near the
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igneous contact to conform more closely to the shape of the reservoir. On the contrary,
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Bergantz and Dawes (1994) argued that given a realistic set of parameters and boundary
conditions, those models considering convection inside the magma chamber do not give
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fundamentally distinctive results from purely conductive ones. Thus, this evidences that
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the influence of the convective processes inside the magma chamber is not well
understood yet and should be analyzed in future works by taking into account as starting
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point, for example, the works by Bea (2010), Gutierrez and Parada (2010) or Koyaguchi
and Taneko (2000).
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Regarding the effects of fluids in the country rock during thermal cooling
processes, conductive heat transfer seems to play the prevailing role in the evolution of
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the aureole (Johnson et al., 2011). However, the country rock permeability and amount of
fluids seem to control the degree to which conduction or convection may dominate the
system (Johnson et al., 2011). The aureoles of intrusions emplaced at shallow crustal
levels are most likely to experience significant heat transfer through fluid convection,
owing to the higher permeability of wall rocks and the greater availability of significant
fluid volumes, including meteoric waters (see Johnson et al., 2011 and references
therein). At depths greater than ~8–10 km, as in our models, the role of fluid convection
diminishes significantly due to decreased permeability and fluid availability (cf. Johnson
et al., 2011)
Probably the most remarkable oversimplification in our models is that the cooling
magma is emplaced instantaneously in the crust, i.e. we are not modeling the
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emplacement and growth of the chamber, but once is already accumulated within the
upper crust. Hence, we are not considering possible thermal effects occurring during
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magma ascent and emplacement. Numerous authors also assumed this simplification
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presumably because of numerical limitations but also due to the fact that the main
mechanisms of magma transport and emplacement into the crust are not fully understood
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(e.g. Gregg et al., 2012;Gottsmann and Odbert, 2014). From diapirism to hydrofracturing
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(dyke injection), there are several theories aimed to explain how magma arrives at
shallow crustal levels and starts accumulating (e.g. Petford and Clemens, 2000; Petford et
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al., 2000). Considering the recent results by Keller et al. (2013), the change from one
mechanism to the other, would mainly depend on the host rock viscosity and tensile
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strength. Numerical models combining transport, emplacement and magma cooling have
been recently developed by Gelman et al. (2013). They simulated formations of large
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magmatic reservoirs by periodic accumulations (controlled by the injection rate) of new
magma pulses at the bottom of the reservoir. Their results allowed to quantitatively
assessing the heat balance between a crystallizing magma and the surrounding crust as it
is periodically rejuvenated by hot recharge from below (cf. Gelman et al., 2013). They
did not focus on how the accumulating sills may thermally affect the country rock, but on
inferring the necessary injection rates to keep melting the upper crust and form huge
(>500 km3) batches of mobile silicic magma. We assume that this aspect will depend on
the sill size, the injection rate and the thermal properties of the crust. Future works are
needed to investigate the thermal effects on the country rock due to magma intrusions
based on the way the latter accumulates in the crust either as large single batch or as
periodic sill injections.
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Finally, our preliminary models allow to broadly constraining the potential
feedback between petrological and mechanical changes. Despite this requires a further
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and detailed study of the endothermic reactions record in the host rock to be incorporated
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into the numerical simulations, our calculated temperature regimes and predicted
petrological changes represent a novel advance on this purpose. As indicated in previous
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works, the incorporation of metamorphic endothermic reactions into the numerical
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simulations decreases the overall temperature throughout the aureole (Cook and
Bowman, 1994) by consuming a significant quantity of heat (Ferry, 1980). This effect
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was investigated in the Cupsuptic pluton (Main, United States) by Bowers et al. (1990)
(Fig. 7). They included the chlorite dehydratation reaction in their numerical simulations,
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which represents the greatest heat sink of all endothermic reactions taking place around
the Cupsuptic pluton. Their results also indicated that
metamorphic
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reaction leads to a thinner contact aureole (Figure 7a, label I) and to slightly different
geometrical distribution of the maximum temperatures around apophyses and pluton’s
geometrical irregularities (Fig. 7a, labels II and III). However, they also showed that the
general
.
the no inclusion in
the numerical simulations of the endothermic reactions may lead to an overestimation of
temperature by as much as 40-50 ºC
. Bowers and co-workers considered a non-
constant latent heat of crystallization that decreases with temperature (Fig. 7c). Therefore,
with time,
our calculated temperature regimes and predicted petrological
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changes (e.g. Fig. 5) have to be considered as maximum estimates. Due to the nature of
the Arrhenius formulation (Eq. 3), to what extent a 40-50 ºC temperature difference
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influences the calculated viscosity for the country rock depends significantly on
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temperature. Thus, for low temperatures, e.g. passing from 400 to 350 ºC, a 50 ºC
disparity would translate in a viscosity increase of nearly one order of magnitude, ranging
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from 2.05  1018 to 1.15  1019 Pa s. However, at higher temperatures, a temperature drop
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of 50ºC (i.e. from 900 to 850 ºC) has minor consequences for the viscosity, which just
increases from 2.20  1013 to 3.81  1013 Pa s. Accordingly, differences in the numerical
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results due to the endothermic reactions will have a lesser effect close to the contact
magma chamber-host rock and at the metamorphic peak, where temperatures are much
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higher.
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Figure 7 (double column):
(a) Results for the thermal conduction model of the
Cupsuptic pluton carried out by Bowers and co-workers (1990). The left figure shows
maximum temperature isotherms computed including effect of the 35 cal/ g of latent heat
released non-linearly according to the crystallization model of Nekvasil as shown in (c).
The right figure includes both latent heat and the effect of the 20 cal/g endothermic
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chlorite dehydration reaction in the aureole. (b) Time-temperature curves at a distance
of 40 m from the contact illustrating the consequences on temperature distribution of
including latent heat of fusion and the biotite isograd endothermic reaction in the
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numerical simulations (c). Curves for latent heat released with temperature decrease for
Cupsuptic melt with 0.25, 0.30, and 0.35 mole fraction water (computed by H. Nekvasil,
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in Bowers et al., 1990). (d) Maximum temperature isotherms obtained from two-
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dimensional conduction heat flow models of an infinite cylinder (left), an infinite
parallelepiped (center) and Cupsuptic intrusion (right) (all figures modified from Bowers
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et al., 1990).
Jellinek and DePaolo (2003) pointed out that viscous deformation of wall rocks may
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influence the dynamics governing the formation of dikes. Their numerical simulations
indicate that this phenomenon is strongly controlled by the influx of new magma into the
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chamber, the reservoir volume, and the country rock rheology. On the one hand,
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chambers of less than 100 km3, with relatively cool wall rocks, can be pressurized up to
the critical overpressure needed to form and propagate dikes from the chamber to the
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surface (i.e. 10-40 MPa) (Fig. 8a). On the contrary, assuming a typical magma supply
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rate, larger reservoirs capable of heating the country rock to a great extent, cannot
achieve this overpressure. Subsequently, there is a strong tendency for magma to
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accumulate in the chamber, which will continue growing in volume as new magma
arrives. Jellinek and DePaolo (2003) stated that it may not be necessary that the viscous
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regime develops around the entire magmatic reservoir, but just with a substantial fraction
of the chamber walls the reservoir would not be able to reach the critical overpressure
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necessary for a magma chamber’s rupture and subsequent dike’s injection. Nonetheless,
their results may vary depending on the magma chamber geometry, e.g. l
ll with higher
curvature (Jellinek and DePaolo, 2003).
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upper crust, such as the one of Rudnick and Gao
(2003). Since the whole magma body and immediate surrounding host rock fall within
the same Bt-Grt-Ab-Kfs-An-Qtz stability field, an increase of the host rock temperature
due to the magma accumulation predicts no mineralogical changes based on phase
diagram modeling (Supplementary Material Figure S1). Thus, while Equation 3 foresees
the existence of a viscous shell around the magma reservoir (Fig. 9a), this may not leave
a visible contact aureole due to petrologic alteration
This relation is a direct result of
the upper crust standard composition used for the P-T pseudosections. While the crustal
composition from Rudnick and Gao (2003) is commonly used in numerical simulations,
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these results indicate that using an average crustal composition may not provide accurate
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petrologic predictions for a local, rapidly changing thermal gradient.
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Figure 7 (single column): Combination of the petrological and thermomechanical
results for the different reservoir geometries. Colored areas correspond to the predicted
mineral assemblage at the metamorphic peak of C for each model (color code as in
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Fig.5), and the magenta solid and the black dashed lines indicate the log10 of the host
rock viscosity (in Pa s) t t=0s (i.e. T = T0_crust , initial crustal temperature based on the
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imposed geothermal gradient) and at the indicated time step , respectively. Roman
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numerals detailed in the text
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AG is grateful for her Ramón y Cajal contract (RYC-2012-11024). A-V thanks the
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assistance of the “Ramón y Cajal” research program (MINECO 2011) and Programa
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Propio I (USal-2014). MD acknowledges the MISTI program (Massachusetts Institute of
Technology [MIT] International Science and Technology Initiative) for funding work at
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the University of Salamanca in 2014. The authors are grateful to Carmen Rodríguez and
an anonymous reviewer for their insightful comments and review of the manuscript,
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which have helped us improve this work. We also thank the Editor Lionel Wilson for his
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comments and for handling this paper.
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